pygae/galgebra

from sympy import symbols, sin, cos, latex
from galgebra.ga import Ga
from galgebra.printer import Format, xpdf

Format()
(x, y, z) = xyz = symbols('x,y,z', real=True)
(o3d, ex, ey, ez) = Ga.build('e_x e_y e_z', g=[1, 1, 1], coords=xyz)
grad = o3d.grad
(u, v) = uv = symbols('u,v', real=True)
(g2d, eu, ev) = Ga.build('e_u e_v', coords=uv)
grad_uv = g2d.grad
A = o3d.lt('A')
print('#3d orthogonal ($A,\\;B$ are linear transformations)')
print('A =', A)
print(r'\f{\operatorname{mat}}{A} =', latex(A.matrix()))
print('\\f{\\det}{A} =', A.det())
print('\\overline{A} =', A.adj())
print('\\f{\\Tr}{A} =', A.tr())
print('\\f{A}{e_x^e_y} =', A(ex ^ ey))
print('\\f{A}{e_x}^\\f{A}{e_y} =', A(ex) ^ A(ey))
B = o3d.lt('B')
print('A + B =', A + B)
print('AB =', A * B)
print('A - B =', A - B)

# FIXME linear transformations fail to simplify
# using dot products of bases

print('#2d general ($A,\\;B$ are linear transformations)')
A2d = g2d.lt('A')
print('A =', A2d)
print('\\f{\\det}{A} =', A2d.det())
# A2d.adj().Fmt(4,'\\overline{A}')
print('\\f{\\Tr}{A} =', A2d.tr())
print('\\f{A}{e_u^e_v} =', A2d(eu ^ ev))
print('\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu) ^ A2d(ev))
B2d = g2d.lt('B')
print('B =', B2d)
print('A + B =', A2d + B2d)
print('AB =', A2d * B2d)
print('A - B =', A2d - B2d)
a = g2d.mv('a', 'vector')
b = g2d.mv('b', 'vector')
print(r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',
      ((a | A2d.adj()(b)) - (b | A2d(a))).simplify())

print('#4d Minkowski spaqce (Space Time)')
m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],
         coords=symbols('t,x,y,z', real=True))
T = m4d.lt('T')
print('g =', m4d.g)
# FIXME incorrect sign for T and T.adj()
print(r'\underline{T} =', T)
print(r'\overline{T} =', T.adj())
# m4d.mv(T.det()).Fmt(4,r'\f{\det}{\underline{T}}')
print(r'\f{\mbox{tr}}{\underline{T}} =', T.tr())
a = m4d.mv('a', 'vector')
b = m4d.mv('b', 'vector')
print(r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',
      ((a | T.adj()(b)) - (b | T(a))).simplify())
xpdf(paper=(10, 12), debug=True)